Hallʹs relations in finite groups

Let H be a finite group, and let θ be an automorphism of H whose order divides n. Hall proved that the number of elements x of H that satisfy the relation xxθxθ2 xθn−1=1 is a multiple of gcd(n,H). For a prime factor p of gcd(n,H), if such a number is not a multiple of gcd(pn,H), then a Sylow p-subgroup of H is exceptional.